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SSC CGL · Cheatsheet

Quantitative Aptitude Quick Revision

Number system, arithmetic, algebra, geometry, mensuration, DI.

Number system basics

Divisibility: 2 (last even), 3 (digit sum ÷3), 4 (last two ÷4), 5 (0/5), 6 (2 & 3), 8 (last three ÷8), 9 (digit sum ÷9), 11 (alt sum diff ÷11).
HCF × LCM = product of two numbers.
Prime numbers up to 100: memorise (25 primes).
Squares 1²..30², cubes 1..15 — learn cold.
1+2+…+n = n(n+1)/2; 1²+…+n² = n(n+1)(2n+1)/6.

Percentage ↔ fraction table

1/2=50%, 1/3=33.33%, 1/4=25%, 1/5=20%, 1/6=16.67%, 1/7=14.28%, 1/8=12.5%, 1/9=11.11%, 1/10=10%.
1/11=9.09%, 1/12=8.33%, 1/13=7.69%, 1/16=6.25%, 1/20=5%, 1/25=4%.
Successive: net = a + b + ab/100 (b sign matters).
Population/price after n years: P(1±r/100)ⁿ.

Profit, loss & discount

Profit%

(SP−CP)/CP × 100

Loss%

(CP−SP)/CP × 100

CP(100+P%)/100

Discount

MP − SP; %=(D/MP)×100

Two disc a,b

single = a+b−ab/100

Tip: If article sold at same SP giving x% gain & x% loss: overall loss = x²/100 %.

Ratio, proportion & partnership

a:b :: c:d ⇒ ad = bc.
Componendo–dividendo: (a+b)/(a−b) = (c+d)/(c−d).
Direct: ↑ together; Inverse: one ↑ other ↓.
Partnership profit share ∝ (capital × time).
Mixture (alligation): ratio = (mean − cheap)/(dear − mean).

Averages & ages

Average = sum/count.
First n natural: (n+1)/2; first n even: n+1; first n odd: n.
Age problems: form equations from 'x years ago/hence'.
Weighted avg = Σ(wᵢxᵢ)/Σwᵢ.

Simple & Compound Interest

PRT/100

CI amt

P(1+R/100)ᵀ

CI 2 yr − SI 2 yr

P(R/100)²

CI 3 yr − SI 3 yr

P(R/100)²(3 + R/100)

Half-yearly CI

P(1+R/200)²ᵀ

Time, speed & distance

Speed

d/t; avg = 2xy/(x+y) equal d

Train + pole

L_train / v

Train + platform

(L_t+L_p)/v

Boat downstream

b+s; upstream b−s

Relative (opp dir)

v₁+v₂

Relative (same dir)

|v₁−v₂|

Tip: 5 km/h = 25/18 m/s; km/h → m/s multiply by 5/18.

Time & work / pipes

1/T

1/T_A + 1/T_B

M₁D₁H₁/W₁

M₂D₂H₂/W₂

Pipe fill/empty

1/T = 1/T_in − 1/T_out

Tip: If A is x times as good as B, A takes 1/x times of B's time; ratio of work = x:1.

Algebra shortcuts

a²−b²

(a+b)(a−b)

a³+b³

(a+b)(a²−ab+b²)

a³−b³

(a−b)(a²+ab+b²)

(a+b)³

a³+3a²b+3ab²+b³

a³+b³+c³−3abc

(a+b+c)(a²+b²+c²−ab−bc−ca)

If a+b+c=0

a³+b³+c³ = 3abc

Geometry — key theorems

Triangle: sum of angles 180°; exterior = sum of two remote interior.
Pythagoras a²+b² = c² for right △.
Similar triangles: sides proportional; area ratio = (side ratio)².
Circle: angle subtended at centre = 2× at circumference (same arc).
Tangent ⟂ radius at point of contact; length from ext point equal.
Cyclic quadrilateral: opposite angles sum 180°.

Mensuration — 2D

Rectangle

A=l×b; P=2(l+b); diag=√(l²+b²)

Square

A=a²; diag=a√2

Triangle

A=½×b×h; equilateral =√3/4 a²

Circle

A=πr²; C=2πr

Sector

A=(θ/360)πr²; l=(θ/360)2πr

Trapezium

A=½(a+b)h

Mensuration — 3D

Cube

V=a³; SA=6a²; diag=a√3

Cuboid

V=lbh; SA=2(lb+bh+hl); diag=√(l²+b²+h²)

Cylinder

V=πr²h; CSA=2πrh; TSA=2πr(r+h)

Cone

V=⅓πr²h; l=√(r²+h²); CSA=πrl; TSA=πr(r+l)

Sphere

V=4/3 πr³; SA=4πr²

Hemisphere

V=2/3 πr³; CSA=2πr²; TSA=3πr²

Trigonometry basics (CGL level)

sin30,45,60,90

1/2, √2/2, √3/2, 1

cos30,45,60,90

√3/2, √2/2, 1/2, 0

tan30,45,60

1/√3, 1, √3

Height & distance

tanθ = opposite/adjacent

Data interpretation

Read chart title, axes, units first.
Note totals; convert % to values only if needed.
Look for 'approximately' → estimate.
Pie chart: 360° = whole; convert degrees ↔ %.
Bar/line: subtract for change; % increase = ((new−old)/old)×100.